Integrate 75 sin3 x cos2 x dx

Due to the spacing between x and its coefficients, I am assuming that the integrand is:
∫ 75sin^3(x)cos^2(x) dx.

If so, note that the power of sin(x) is odd and the power of cos(x) is even, so we need to pull out a factor of sin(x) from the integrand and "absorb" it into the dx term as follows:
∫ 75sin^3(x)cos^2(x) dx = 75 ∫ sin(x)sin^2(x)cos^2(x) dx
= 75 ∫ sin^2(x)cos^2(x) [sin(x) dx]
= 75 ∫ cos^2(x)[1 - cos^2(x)] [sin(x) dx], by the Pythagorean Identities.

Now, if we apply the substitution u = cos(x) ==> du = -sin(x) and -du = sin(x) dx:
75 ∫ cos^2(x)[1 - cos^2(x)] [sin(x) dx] = 75 ∫ u^2(1 - u^2)(-1) du
= 75 ∫ u^2(u^2 - 1) du, since -(1 - u^2) = -1 + u^2 = u^2 - 1
= 75 ∫ (u^4 - u^2) du, by multiplying through
= 75[(1/5)u^5 - (1/3)u^3] + C, by integrating with the Power Rule
= 15u^5 - 25u^3 + C, by multiplying the 75 through.

Therefore, back-substituting u = cos(x) yields:
∫ 75sin^3(x)cos^2(x) dx = 15cos^5(x) - 25cos^3(x) + C.

I hope this helps!

Is that:

sin(3x) * cos(2x) * dx

or

sin(x)^3 * cos(x)^2 * dx


Either way, I'll show you


int(75 * sin(3x) * cos(2x) * dx) =>
75 * int(sin(3x) * cos(2x) * dx) =>
75 * int((1/2) * (2 * sin(3x) * cos(2x) + sin(2x) * cos(3x) - sin(2x) * cos(3x)) * dx) =>
75 * (1/2) * int(sin(2x)cos(3x) + sin(3x)cos(2x) + sin(3x)cos(2x) - sin(2x)cos(3x)) * dx) =>
(75/2) * int((sin(2x + 3x) + sin(3x - 2x)) * dx) =>
(75/2) * int((sin(5x) + sin(x)) * dx) =>
(75/2) * int(sin(5x) * dx) + (75/2) * int(sin(x) * dx) =>
(75/2) * (-1/5) * cos(5x) + (75/2) * (-1) * cos(x) + C =>
(75/2) * (-1/5) * (cos(5x) + 5 * cos(x)) + C =>
(-15/2) * (5 * cos(x) + cos(5x)) + C



int(75 * sin(x)^3 * cos(x)^2 * dx) =>
75 * int(sin(x)^2 * sin(x) * cos(x)^2 * dx) =>
75 * int((1 - cos(x)^2) * sin(x) * cos(x)^2 * dx)

u = cos(x)
du = -sin(x) * dx

75 * int((1 - u^2) * (-du) * u^2) =>
-75 * int((u^2 - u^4) * du) =>
-75 * ((1/3) * u^3 - (1/5) * u^5) + C =>
-75 * (1/15) * u^3 * (5 - 3u^2) + C =>
-5 * cos(x)^3 * (5 - 3 * cos(x)^2) + C

∫75sin^3(x) * cos^2(x) dx

75∫sin(x)*(1 - cos^2(x))(cos^2(x)) dx

z = cos(x) => -dz = sin(x) dx

75*∫(z^2 - 1)(z^2) dz

= 15z^5 - 25z^3 + C

= 15cos^5(x) - 25cos^3(x) + C

= cos^3(x) * (15cos^2(x) - 25) + C